Final answer:
To find the derivative of the function y = x⁴ cos(x) using logarithmic differentiation, you can take the natural logarithm of both sides of the equation, apply the properties of logarithms, and differentiate using the chain rule.
Step-by-step explanation:
To find the derivative of the function y = x⁴ cos(x) using logarithmic differentiation, we will start by taking the natural logarithm of both sides of the equation. This will allow us to use the properties of logarithms to simplify the expression.
ln(y) = ln(x⁴ cos(x))
Next, we will use the properties of logarithms to simplify the expression further. The logarithm of a product is equal to the sum of the logarithms of the factors, and the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.
ln(y) = 4ln(x) + ln(cos(x))
Now, we will differentiate both sides of the equation with respect to x using the chain rule.
(1/y)(dy/dx) = 4(1/x) - tan(x)
Finally, we can solve for dy/dx by multiplying both sides of the equation by y.
dy/dx = y(4/x - tan(x))
Substituting back the original expression for y, we get the derivative of the function:
dy/dx = x⁴ cos(x)(4/x - tan(x))